Dimensions of locally and asymptotically self-similar spaces

By S. Buyalo, Nina Lebedeva. 2005

56 citations recorded by the source corpus.

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdimG = dim@1G + 1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X ×Y ) < asdimX+asdimY . Other applications are also given.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/23976b3ce755e00e18e766eea7c0e6f207a1fea2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The asymptotic dimension of a curve graph is finite (2005, background): “For finitely generated hyperbolic groups Γ, it is known that dim ∂Γ + 1 = asdim Γ by results of Buyalo and Lebedeva [11]; cf. [12, 20, 42].”
Embedding of hyperbolic groups into products of binary trees (2007, methodology): “Remark 8.1. In [Bu, BL ], this notion is called the capacity dimension.”
Dimension zero at all scales (2006, unknown): “For bounded metric spaces the notion of Assouad-Nagata dimension is equivalent to the notion of capacity dimension introduced recently by Buyalo [4] [5].”